Fill a high-sided pan with water and heat it with vinegar until the first signs of boiling.
Ramp up the liquid with a spatula to create a vortex in the centre and cook the poached eggs, one at a time, for three minutes.
Drain the eggs with a skimmer and let them cool for 2 minutes in very cold water. Dry them on absorbent paper and keep them aside.
Heat the cream in a saucepan and reduce it for about 10 minutes, adjusting it with salt and pepper.
Finely chop the well washed and dried mint and cut the mild Provolone Valpadana P.D.O. into 4 1 cm thick washers. Scratch the rind of the two lemons and keep it aside.
Place 2 eggs in each cocotte, cover with cream, sprinkle with mint and put the cheese washers on top.
Bake in grill mode, at 220°, about 4 minutes, just enough time to gratin the mild Provolone Valpadana P.D.O. cheese.